# The linear arboricity of graphs

- Received:

DOI: 10.1007/BF02783300

- Cite this article as:
- Alon, N. Israel J. Math. (1988) 62: 311. doi:10.1007/BF02783300

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## Abstract

A*linear forest* is a forest in which each connected component is a path. The*linear arboricity* la(*G*) of a graph*G* is the minimum number of linear forests whose union is the set of all edges of*G*. The*linear arboricity conjecture* asserts that for every simple graph*G* with maximum degree Δ=Δ(*G*),\(la(G) \leqq [\frac{{\Delta + 1}}{2}].\). Although this conjecture received a considerable amount of attention, it has been proved only for Δ≦6, Δ=8 and Δ=10, and the best known general upper bound for la(*G*) is la(*G*)≦⌈3Δ/5⌉ for even Δ and la(*G*)≦⌈(3Δ+2)/5⌉ for odd Δ. Here we prove that for every*ɛ*>0 there is a Δ_{0}=Δ_{0}(*ɛ*) so that la(*G*)≦(1/2+*ɛ*)Δ for every*G* with maximum degree Δ≧Δ_{0}. To do this, we first prove the conjecture for every*G* with an even maximum degree Δ and with*girth g*≧50Δ.